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Co-design of the LCL Filter and Control for Grid-Connected Inverters
Zhang, Yu; Xue, Mingyu; Li, Minying; Kang, Yong; Guerrero, Josep M.
Published in:
Journal of Power Electronics
DOI (link to publication from Publisher):
10.6113/JPE.2014.14.5.1047
Publication date:
2014
Document Version
Early version, also known as pre-print
Link to publication from Aalborg University
Citation for published version (APA):
Zhang, Y., Xue, M., Li, M., Kang, Y., & Guerrero, J. M. (2014). Co-design of the LCL Filter and Control for GridConnected Inverters. Journal of Power Electronics, 14(5), 1047-1056. DOI: 10.6113/JPE.2014.14.5.1047
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Co-design of LCL Filter and Control for Grid-Connected Inverters
1
http://dx.doi.org/10.6113/JPE.2012.12.4.000
JPE 14-03-129
Co-design of the LCL Filter and Control for
Grid-Connected Inverters
Yu Zhang*, Mingyu Xue*, Minying Li**, Yong Kang* and Josep M. Guerrero†
*
State Key Laboratory of Advanced Electromagnetic Engineering and Technology (AEET), Huazhong University of
Science and Technology, Wuhan, China
**
Guangdong Zhicheng Champion Group Company, Dongguan, China
†
Microgrids research programme: www.microgrids.et.aau.dk email: [email protected]
Department of Energy Technology, Aalborg University, Aalborg, Denmark
Abstract
In most grid-connected inverters (GCI) with an LCL filter, since the design of both the LCL filter and the controller is done
separately, considerable tuning efforts have to be exerted when compared to inverters using an L filter. Consequently, an integrated
co-design of the filter and the controller for an LCL-type GCI is proposed in this paper. The control strategy includes only a PI
current controller and a proportional grid voltage feed-forward controller. The capacitor is removed from the LCL filer and the
design procedure starts from an L-type GCI with a PI current controller. After the PI controller has been settled, the capacitor is
added back to the filter. Hence, it introduces a resonance frequency, which is identified based on the crossover frequencies to
accommodate the preset PI controller. Using the proposed co-design method, harmonic standards are satisfied and other practical
constraints are met. Furthermore, the grid voltage feed-forward control can bring an inherent damping characteristic. In such a way,
the good control performance offered by the original L-type GCI and the sharp harmonic attenuation offered by the latter designed
LCL filter can be well integrated. Moreover, only the grid current and grid voltage are sensed. Simulation and experimental results
verify the feasibility of the proposed design methodology.
Key words: Grid-connected inverter, LCL filter, Current control, Grid voltage feed-forward control, Resonance damping, Co-design
I.
INTRODUCTION
Grid-connected inverters (GCIs) are widely used in
distributed generation (DG) systems where the sources are
mostly renewable energies like wind, solar, etc. The traditional
L filter is gradually being replaced by LCL filters, due to their
stronger harmonic elimination capability [1][2].
A PI controller is a simple robust controller that can be
effectively used for current control in GCIs with an L filter.
This is due to the fact that PI controller are easy to design to
achieve good performance. For instance, in [3], a PI controller
was designed to achieve a maximum crossover frequency
while considering the control delay and the PWM transport
delay. When combined with proportional grid voltage
feed-forward control, grid disturbances can also be rejected
effectively.
However, the design of PI controller for an LCL-type GCI is
Manuscript received Mar. 26, 2014; revised June 4, 2014
Recommended for publication by Associate Editor ----.
*Corresponding Author: [email protected]
Tel: +86-2787543658-810, Fax: +86-2787543658-816,
School of Electrical and Electronic Eng., Huazhong University of Science
and Technology, China
not easy due to the resonance of the LCL filter, which may lead
to instability and a complex design in terms of the grid current
feed-back controllers and grid voltage feed-forward controllers
[4]-[6]. With the conventional design method, using PI current
control is not enough to achieve stability. In [7], the regions of
the active damping control decided by the resonance frequency
were discussed. It was concluded that active damping was
needed for lower resonance frequencies. Most of the proposed
active damping strategies are realized by using multi-variable
feedback controls [8]-[12]. Some of them may also lower the
system order by using multi-variable feedback controls
[13]-[15]. However, multi-variable feedback control lifts the
cost of the sensors and makes the control strategy more
complicated. Therefore, in [4], a filter-based active damping
solution is proposed. In [16], it is concluded that when
converter-side current is used as a control feedback variable,
there is an inherent damping term embedded in the control loop,
which can neutralize the resonance introduced by an LCL-filter.
This damping term can also be obtained when grid
feed-forward is implemented, which will be discussed in this
paper.
2
Journal of Power Electronics, Vol. 14, No. X, July 2014
When the resonance frequency increases, the phenomenon
of the magnitude curve crossing over 0 dB three times can be
observed in the Bode diagram of an LCL-type GCI with a PI
controller. Hence, according to the Nyquist Stability Criterion
[17], three crossover frequencies should be considered together
to confirm the stability. However, this has not been discussed
in the literature. To accurately ensure stability, three cross over
frequencies will be discussed in this paper. They are considered
together in the design according to the Nyquist Stability
Criterion.
In the conventional designs of an LCL-type GCI, the design
of both the LCL filter and the controller is done separately.
They always start with the parameter design of the LCL filter,
which aims to satisfy switching current harmonic standards and
other practical constraints [18]–[20]. Thereafter, the controller
is added and designed according to this preset LCL filter. For
the LCL filter, various optimization goals are pursued, such as
minimum reactive power [21], low volume, low loss [22], and
highest power efficiency [23]. In some converters, the
controller is considered during the design of the filter. In [24], a
generalized design method of an LCL-type active power filter
is proposed, with a capacitor current feedback control loop
being used for active damping. In [25], the LC filter of the
inverter is designed when considering the performance of the
close-loop system. However, among the conventional methods
of GCI design, most of the power stage filters and controllers
are designed separately. Hence, LCL-type GCIs suffer from
complexity in terms of hardware and software, and are hard to
properly tune. Sometimes poor performance is achieved in
comparison to the L-type GCIs.
In DG systems, the primary task of energy conversion
requires a further simplification such as a lower number of
variables to be sensed. As required by industry applications,
GCIs should behave well in terms of current reference tracking,
grid disturbance rejecting, and robustness to grid impedance
variations. All of these are well featured by L-type GCIs.
Hence, a novel co-design method for an LCL-type GCI and its
controller is proposed in this paper to achieve all of these
purposes. This is theoretically feasible based on multi
crossover frequencies according to the Nyquist Stability
Criterion.
In the first step of the co-design method proposed in this
paper, the capacitor is removed from the LCL filter. This
converts it to an L type filter, so that the design starts from an
L-type GCI employing a PI controller. After the PI controller
has been set for the L filter, the capacitor is added back to the L
filter. This will introduce a resonance frequency to the LCL
filter, which will be adapted to the preset PI controller by
identifying the phase margin of three crossover frequencies.
Furthermore, proportional feed-forward control of the grid
voltage is employed to ensure performance under a highly
distorted or weak grid. This brings an inherent damping
characteristic. By an easy method, the LCL filter and controller
can be co-designed, and both good close-loop performance and
sharp harmonic attenuation can be achieved. Moreover, only
the output current and grid voltage are sensed. Simulation and
experiment results verify the feasibility of the proposed
co-design method.
II.
CONCEPT OF THE PROPOSED CO-DESIGN
The power stage model and controller of a GCI with an
LCL filter is shown in Fig. 1(a), where the dc link can be
simplified as a constant voltage U dc if the dc capacitor Cd
is large enough. Resistors r1, r2 and rg can be neglected, while
the grid internal impedance Lg is considered [16]. The
corresponding control scheme is presented in Fig. 1(b), where
GPI ( s ) is the PI current controller and G ff ( s ) is the grid
voltage feed-forward controller.
When the capacitor C f is removed from the LCL filter, the
filter becomes an L type, and L= L1 + L2 . In Fig. 1(b), this
means to “short-circuit” the dashed block in the power stage
model. The e- sTd term represents the system delay, and Td
is the one-sample control delay plus the half-sample PWM
transport delay [3].
For an L-type GCI, a PI current controller together with a
unity proportional feed-forward controller for the grid voltage
is commonly employed for current reference tracking and grid
voltage disturbance rejection. The state of the art technique for
designing a PI current controller is based on maximizing the
proportional part and minimizing the integral time constant to
obtain the maximum crossover frequency. This results in a
good coordination between the dynamics and the steady-state.
+
Q1
D1
Q3
D3
( L1 + L2 ) + ( r1 + r2 ) =L + r



r1
L1
i1
i2
M
U dc
u PWM
Cf
Q2
D2
rg
r2
+
uCf
ug
-
-
-
Cd
L2
+
+
Lg
Us
Grid
Q4
D4
-
(a) Topology
ug
Controller
Power Stage Model
C f L1s 2 + 1
G ff ( s )
i2*
GPI ( s )
+
d
U dc ⋅ e − sTd
u PWM
-
1
( L1 + L2 ) s
(b) Control models
Fig. 1. Single-phase GCI with LCL filter.
ωr2
s 2 + ωr2
i2
3
Co-design of LCL Filter and Control for Grid-Connected Inverters
In detail, a PI controller GPI =
( s ) k P (1 + 1 τ s ) for an
L-type GCI is governed by: k P = ωc (max) L / U dc and
τ =10 ωc (max) , where ωc (max) represents the maximum cross
over frequency that determines the expected phase margin
φm [3].
As a result of this design, the open-loop transfer function
from the duty-cycle d to the output-current i2 is shown as:
GLo ( s )
=
ω
i2 ( s ) U dc GPI ( s )e
e
=
=
ωc (max) (1 + c (max) )
d (s)
( L1 + L2 ) s
10 s
s
− sTd
− sTd
0.4
0.2
Pcrit
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1
-0.8
-0.6
(1)
Fig. 2(a) shows an open-loop Nyquist diagram of an L-type
GCI after the design. GCIs with an L filter have a low
harmonic attenuation capability. Hence, they can be improved
by an LCL filter, which can be realized by returning the filter
capacitor C f to the splitting point M of the L filter, as
shown by the dashed block in Fig. 1(a). For the LCL structure,
the open-loop transfer function from the duty-cycle to the
output-current is given by:
i2 ( s ) U dc GPI ( s )e − sTd
ω2
1
GLoC=
GLo ( s ) 2 r 2
=
=
L (s)
2
2
d (s)
L1 L2 C f s
s + ωr
s + ωr
(a)
It can be seen that in the case of a lower ωr , pu = 1 (see
line 1), there is only one negative crossing of 0 dB in the
-0.2
0
0.2
0.4
L-type GCI
12
10
8
6
4
2
Pcrit
O
0
-2
-4
-15
-10
(2)
here, =
( L1 + L2 ) / L1 L2 C f is the resonance frequency
ωr
of the LCL filter.
With the PI controller GPI ( s ) unchanged and GLo ( s )
remaining invariant with a total inductance L= L1 + L2 , the
stability after upgrading relies on the value of the resonance
frequency ωr . Based on (2), Fig. 2(b)-(c) illustrates the open
loop Nyquist diagrams of LCL-type GCIs with three different
filter resonance frequencies, which are represented in p.u.
values, with the sampling frequency 1 T as the base
frequency, i.e. ωr , pu = ωr T . These Nyquist curves show the
loci determined by a positive ω from 0 to infinity. The
radius of the semicircles in (b)-(d) is infinity, which is the
result of the filter resonance frequency. Pcrit denotes the
critical point: -1+j0.
It can be estimated from (2) that the Nyquist curve will
cross the negative real axis from the left of the critical point
Pcrit when ωr , pu < 1.06 , which means instability. However, it
will cross the negative real axis from the right of Pcrit when
ωr , pu > 1.06 , which means stability. Furthermore, if ωr , pu
is increased to 9, the GCI will return to instability. Therefore,
the determination of ωr , pu is very important in the design.
It is convenient to further discuss the relative stability in
the Bode diagram. Fig. 3 illustrates a Bode diagram through
three different filter resonance frequencies (see lines 1, 2, 3).
-0.4
-5
0
5
10
15
LCL-type with ωr , pu = 1
(b)
8
6
4
2
Pcrit
O
0
-2
-4
-6
-8
-4
-2
0
(c)
2
4
6
8
LCL-type with ωr , pu = 1.74
5
4
3
2
1
Pcrit
0
O
-1
-2
-3
-4
-5
-4
-3
-2
-1
0
1
2
3
4
5
(d) LCL-type with ωr , pu = 2.5
Fig. 2. Open-loop Nyquist diagram of L-type and LCL-type
GCI.
4
Journal of Power Electronics, Vol. 14, No. X, July 2014
50
2
Magnitude (dB)
1
3
d
0
e
L-GCI
-50
-90
b
Phase (deg)
-180
-270
a
-360
-450
-540
c
ωc1 104
ω- π
Frequency (rad/sec)
ωc 2
ωc 3
Fig. 3. Open-loop Bode diagram of LCL-type GCI against
ωr , pu with L-type counterpart (dashed line) as the benchmark.
(Line 1: ωr , pu = 1 ; line 2: ωr , pu = 1.74 ; line 3: ωr , pu = 2.5 )
whole frequency range. In addition, the phase of the
crossover frequency is lower than − π . This kind of crossing
may lead to instability. Note that, as ωr , pu increases, three
crossover frequencies will appear. The phase angle of the
lowest crossover frequency will become larger than − π ,
which could possibly ensure stability (see lines 2 and 3). This
phenomenon can help design a controller more effectively,
which means that the three crossover frequencies need to be
identified and made to have enough phase margin by properly
fixing the resonance frequency ωr . This can ensure the
stability and performance of the system more accurately.
III. DESIGN PROCEDURE
The aim of this section is to present a systematic approach
to select the main control parameters for the LCL-type GCIs,
by taking the L-type control design as a first step.
A. Identify the crossover frequencies of an LCL filter to
adjust the PI controller
ωr is increased and three crossover frequencies are made
to appear. According to the Nyquist Stability Criterion, it is
possible to ensure the stability and performance of systems
by making them with enough of a phase margin. Fig. 3 shows
the three crossover frequencies of line 2, named ωc1 , ωc 2 ,
and ωc 3 in ascending frequency order. Their corresponding
phase angles are ϕ1 =− π + φm1 , ϕ 2 =− π − φm 2 and
ϕ3 =
−3π + φm 3 , where φm1 , φm 2 and φm 3 represent the
phase margins [17]. In addition, the gain margin is measured
at a phase angle of − π , and the corresponding frequency is
defined as ω-π . In Fig. 3, the points ‘a’, ‘b’, and ‘c’ denote
the phase margin for line 2, while point ‘d’ denotes its gain
margin, and point ‘e’ denotes the gain margin of line 3.
When the capacitor C f is removed from the LCL filter in
Fig. 1, it becomes an L filter with L= L1 + L2 . The PI
current controller can be designed easily because a
comparatively large phase margin can be reserved, which is
set as φm = 55° here. The integral term of the PI controller
reduces the phase margin, i.e. about 5° . Hence, for an L-type
GCI with a sample interval of T=50 μs , the phase margin
φm can be set as 49.1º and the gain margin λL can be set as
7.97 dB. According to these constrains, the parameters of the
PI controller can be optioned to get the maximum band width
ωc ( max ) [3].
The capacitor C f then returns and subsequently the L
filter changes back to an LCL filter. It can be determined by a
numerical evaluation that if ωr approaches infinity, the
stability margin will be determined by the gain margin at
frequency ω-π , which is the same as its L-type counterpart
(see Fig. 3). Assuming a gain margin λLCL for the LCL-type
GCI (it should be smaller than its L-type counterpart for the
sake of feasibility) and referring to (2), the lower limit of
ωr _ low can be determined by solving:
λL − λLCL =20 ⋅ log10 (
ωr2_ low
)
ωr2_ low − ω-π2
(3)
Although the gain margin increases with ωr , the phase
margin φm 3 decreases. Then from (2), the upper limit of
ωr _ up can be found as follows:
 ωc ( max )
ωr2_ up
·
≈ -1

2
2
 ωc 3 ωr _ up − ωc 3

arctan( 10ωc 3 ) − ω T − 2π =
−3π + φm 3
c3 d

ωc ( max )

(4)
Although the stability is guaranteed, there is still a
resonance peak, which may cause problems when the GCI is
aiming to track current harmonics around the resonance
frequency. However, in DG systems where GCIs are mostly
in charge of energy conversion, the resonance peak should
not be troublesome.
B. Parameters design of the LCL filter
Once the range of the resonance frequency ωr has been
identified, the parameters of the LCL filter should be
determined. It is well known that the larger the total
inductance of the filter, the larger the fundamental voltage
drop. This also increases the amount of reactive power that
can be absolved. On the other hand, for low values of the
inductance, a high current ripple is obtained, which results in
higher iron losses and more current stress on the
semiconductors. Therefore, constraints on the inductances
should be set as a priority. The preliminary parameters of the
5
Co-design of LCL Filter and Control for Grid-Connected Inverters
LCL filter, including the resonance frequency ( ωr ), the
maximum allowable grid current ripple in a percentage of the
rated value ( %
i2 = ∆i2 I 2,n ), and the inductor split factor
( k L = L2 / L1 ), are selected by using the following steps:
1) Design of %
i2 : A lower value of %
i2 requires a larger total
required inductance. According to IEEE std. 519-1992
[28], the upper limit is 30% without a loss of generality.
The value of %
i2 can be initialized to 25% and then
adjusted according to the amount of total inductance;
2) Design of ωr : Given %
i2 , a larger value of the total
inductance requires a higher ωr [18]. Therefore, ωr
should be set as lower as possible, e.g. the lower
limitation required by the PI controller;
3) Design of k L : The inverter current ripple reaches its
maximum at k L = 1 . Therefore, k L should be set as a
trade-off
between
the
current
ripple
(e.g.,
%
i1 =
∆i1 I 2,n ∈ [15%, 40%]) and the reactive power (e.g.,
<5%). A good trade-off can be achieved by selecting
k L = 0.3 .
After that, the values of the filter components can be
determined. The harmonic in u PWM corresponding to the
maximum carrier sideband harmonic of the grid current can
be identified by applying a Bessel function. In terms of an
H-bridge with a Unipolar SPWM, the (2N+1)th order
harmonic current (N is the carrier ratio) represents the most
significant component [29]. The amplitude of the
corresponding harmonic voltage is governed by:
2U dc
(5)
Uˆ1,1 =
J1 (Mπ)
The grid-side inductance and filter capacitance can be
selected as:
(8)
L2 = k L L1
and:
Cf =
L1 + L2
L1 L2ωr2
(9)
IV. DESIGN EXAMPLE WITH PERFORMANCE
EVALUATION
According to the above procedure, a design example of a
2.5 kW GCI with an LCL filter is presented in Table I, where
the total inductance is 0.025 p.u., the inverter current ripple is
25%, and the capacitive power is 0.02 p.u. Consequently, the
presented design fits well with the practical constraints. Then,
the evaluation of control performance is carried out as
follows.
A. Reference tracking
The load step response of both the L-type and the
LCL-type filters are presented in Fig. 4. They show a similar
bandwidth but a higher overshoot that lasts for about 2T.
However, such a peak only occurs in the averaged model.
Hence, it can be used to check the stability margins. During
the initial overshoot, the averaged model is not valid. As a
result, a more smooth dynamics is available, as proved in
Section V.
TABLE I
PARAMETERS OF LCL FILTER AND REACTIVE COMPONENT
π
U dc
378 V
here, the subscript of Û1,1 represents the first sideband of the
first carrier band, and J1 is a first order Bessel function. M
is the modulation depth. M varies little with load changes.
Therefore, it can be approximated by the modulation depth
for the no-load condition, i.e. M = uˆ g U dc , where uˆ g is
the magnitude of u g .
Back to Fig. 1, the output current can be determined as
follows:
ug
220 Vrms
I 2,n
11.5 Arms
L1
1.2 mH
L2
0.35 mH
Cf
3.3 μF
ωr , pu
1.64
%
i2
25%
i2 ( s )
ω
2
r
s ( L1 + L2 )( s + ω )
2
2
r
[uPWM ( s ) − ( L1C f s 2 + 1) ⋅ u g ( s )]
(6)
When uPWM ( s ) = Uˆ1,1 and u g ( s ) = 0 , the inverter-side
inductance can be set as:
L1
=
Uˆ1,1
ωr2
⋅ 2
2
i2 I 2, n (1 + k L )ω2 N +1 ω2 N +1 − ωr
2 ⋅%
(7)
where ω2 N +1 is the angular frequency of the (2N+1)th order
harmonic.
B. Rejection of grid disturbances
The inverter output impedance is defined as:
Z o ( s ) = ug ( s ) i2 ( s )
(10)
The larger the value of Z o , the stronger the rejection of
grid voltage disturbances [10] [26]. Hence, if the output
impedance is high enough in the whole frequency range, the
power quality will be much more improved in terms of
6
Journal of Power Electronics, Vol. 14, No. X, July 2014
may not be the best solution.
1.6
LCL
L
1.4
C. Inherent damping characteristic from the
feed-forward control of grid voltage
1.2
When unity grid voltage feed-forward control is used, the
grid voltage u g is measured at the point of common
coupling (PCC). Assuming weak grid conditions,
i.e., Lg = nL2 , the grid voltage can be expressed as follows:
n
1
(12)
=
ug
uCf +
uS
1+ n
1+ n
The open-loop transfer function of an LCL-type GCI with
the grid voltage feed-forward control can be expressed as:
Amplitude
1
0.8
0.6
0.4
0.2
0
0
Fig. 4.
0.5
1
1.5
2
Time/sec
2.5
3
3.5
4
-3
x 10
Step responses of L-GCI and LCL-GCI.
harmonic distortion. Fig. 1(b) shows that if an exact full grid
voltage feed-forward control is employed:
G=
( L1C f s 2 + 1) ⋅ e sTd
ff ( s )
(11)
The output impedance tends to be infinite. However, the
second-order differentiator and the lead phase blocks can be
neglected. Thus, it is possible to approximate e sTd ≈ 1 and
L1Cfs2=1, so that G ff ( s ) = 1 is still preferred. With these
assumptions, the Bode diagram is illustrated in Fig. 5. It
shows a lot of improvement in the output impedance
(compare line 2 and line 1), which is only little degraded by
the grid impedance. It is worth mentioning that if
multi-feedback active damping control is employed (e.g.,
capacitor current feedback [27]), the subsequent negative
impact from the grid impedance will be noticeable. As a
result, the traditional unity feed-forward control G ff ( s ) = 1
where ωr , g =


nk L
s 2 + ωr2, g 1 − G ff ( s ) ⋅
e − sTd 
1
+
(
1
+
n
)
k

L

(13)
L1 + L2 + Lg
.
L1 ( L2 + Lg )C
While the open-loop transfer function of the corresponding
L-type counterpart can be expressed as:
U ⋅ G ( s ) ⋅ e − sTd
(14)
GLo, g ( s ) = dc PI
LT , g s
where LT , g = L1 + L2 + Lg .
The delay block e − sTd can be approximated by its first
− sT
order Taylor series, i.e. e d ≈ 1 − Td ⋅ s . Hence, (13) can be
expressed as:
o
o
GLCL
, g ≈ GLCL , g
where K D g ff
=
140
ωr , g 2
(15)
s 2 + K D Tdωr , g 2 s + (1 − K D ) ⋅ ωr , g 2
nk L
∈ (0,1) .
1 + (1 + n)k L
50
Magnitude (dB)
120
100
Magnitude (dB)
ωr2, g
o
o
GLCL
, g ( s ) = GL , g ( s ) ⋅
80
2
2
1
0
3
-50
-90
60
1
-180
Phase (deg)
3
40
20
-270
1
3
-360
2
-450
-540
0
0
10
1
10
50
2
10 150
3
10
Frequency (Hz)
Fig. 5. Output impedance of LCL-GCI:
(Line 1: Lg = 0 , G ff ( s ) = 0 ; line 2: Lg = 0 , G ff ( s ) = 1 ;
line 3: Lg = 3L2 , G ff ( s ) = 1 )
4
10
-630
2
10
3
10
Frequency (Hz)
4
10
Fig. 6. Open-loop Bode diagram against grid impedance.
(1. Lg = 0 , G ff ( s ) = 0 ; 2. Lg = 3L2 , G ff ( s ) = 0 ;
3. Lg = 3L2 , G ff ( s ) = 1 ).
7
Co-design of LCL Filter and Control for Grid-Connected Inverters
From (15), it can be observed that if G ff ( s ) is a positive
constant g ff , a damping coefficient ζ is introduced due to
the phase delay introduced by e − sTd . This can be expressed
as:
1 KD
(16)
ζ =
ωr , g Td
2 1− KD
As illustrated in Fig. 6, the unity feed-forward under a
weak grid not only preserves the stability margin, but also
makes the loop gain almost unchanged (compare line 3 with
line 2). Compared with the existing active damping methods,
where the bandwidth is mostly reduced by negative feedback
of state variables, the proposed approach gives the active
damping benefit without any additional control loop.
Load step-down instant
Load step-up instant
V /V
500
g
0
u
(
V
)
g
-500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.02
0.03
0.04
0.05
time/sec
Time
(sec)
0.06
0.07
0.08
I /A
20
1
0
(A
)
1i
-20
0.01
20
(
A
)
I /A
V.
SIMULATION AND EXPERIMENTAL RESULTS
A. Simulation results
A Matlab/Simulink model is developed to verify the
proposed integral design of the GCI together with the LCL
filter, using the parameters listed in Table I. The harmonics
content shown in Table II are introduced to the grid voltage.
Load step changes from no-load to full-load and vice versa
are performed under weak grid condition with different grid
impedances.
The waveforms shown in Fig. 7(a) illustrate that by using a
PI controller, the grid current is severely distorted, and the
compatibility of the grid impedance is poor, due to the
resonance at the load switching instant if Lg = L2 . However,
such a condition is greatly improved by the grid voltage
feed-forward control, as shown in Fig. 7(b), where Lg = 3L2 .
Moreover, the system dynamics are smoother than the
averaging model based analysis (see Fig. 4, which has stiff
grid conditions), due to the strong proportional part and the
comparatively weak integral part of the PI controller.
Actually, the LCL-type GCI behaves similar to the L-type
one, except for the sharp harmonic attenuation, as detailed in
Fig. 7(c), where the switching grid current under stiff grid
conditions is about 0.26% of the rated current.
2
0
TABLE II
2i
-20
0.01
COMPONENTS OF GRID HARMONICS
(a) G ff ( s ) = 0 , Lg = L2
Load step-up instant
3rd
5th
7th
9th
Load step-down instant
500
u
(V
)
V /V
g
0
4.82%
4.18%
3.54%
2.9%
g
-500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
B. Experimental results
0.08
o
rB
20
(
A
)
I /A
1
0
ard
1i
-20
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Po
20
oard
ard
0
2
rol B
o
eB
iv
Dr
2i
-20
0.02
0.03
0.04
0.05
time/sec
Time
(sec)
0.06
0.07
0.08
or
s
0.01
ns
(A
)
I /A
Cont
we
Se
(b) G ff ( s ) = 1 , Lg = 3L2
THD = 7.52%
5
I1,h/I1,n(%)
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Harmonic order
THD = 2.27%
Inductors
I2,h/I2,n(%)
0.9
0.6
0.3
0
Fig. 8. Photograph of the single phase GCI prototype.
0
50
100
150
200
250
300
350
400
Harmonic order
(c) Harmonic spectrum. (Subscripts h and n denote harmonics
and fundamental components respectively.)
Fig. 7.
Simulations under weak and stiff grid conditions
8
Journal of Power Electronics, Vol. 14, No. X, July 2014
The proposed design strategy is experimentally verified on
a 2.5 kW prototype, as shown in Fig. 8.
The dc and ac buses are formed by an isolated dc source
and the utility grid, respectively. The control is implemented
by a 32 bit fixed-point 100 MHz TMS320F2808 DSP.
Unipolar-SPWM is used, the switching frequency is 10 kHz
and the sampling frequency is 20 kHz. Since there is no need
for active damping, only the grid current and the grid voltage
are sensed. The sampling action is taken at the top and
bottom of the carrier signal. The dead time for the power
switches is set to 1 μs. The parameters are selected as shown
in Table I.
Firstly, under stiff grid conditions, load steps from no-load
to full-load, and vice versa, are performed. The resulting
waveforms of the inverter side current and grid side current
are shown in Fig. 9. They show smooth and fast dynamics
and good power quality. The THD of the grid current is well
below the standard requirements, as shown in Fig. 9(b) (i.e.
2.78% vs. 5%). Notice that there is some deviation that is
mostly accounted for by the measurement susceptible to EMI
during IGBT switching, and reflected by the harmonics of the
200th order that should theoretically be null. Note that %
i2 is
about 25%, implying that the dominant iron loss is low.
Fig. 10. Experimental waveforms of grid voltage and
current in weak grid (Lg=1mH)
Then, load steps under weak grid conditions are emulated
by inserting another inductor Lg = 1 mH between the PCC
and the grid. The waveforms shown in Fig. 10 illustrate the
good behavior of the system even for large grid impedance
values. Moreover, there is no obvious deterioration of the
power quality. This is ensured by the large output impedance
and the preserved loop gain (see Fig. 5 and Fig. 6).
VI. CONCLUSIONS
(a) G ff ( s ) = 0 , Lg = L2
THD = 7.04067 %
I1,h/I1,n(%)
4
3
2
1
0
0
50
100
150
200
250
300
350
400
250
300
350
400
Harmonic order
THD = 2.78002 %
1.8
I2,h/I2,n(%)
1.5
1.2
0.9
0.6
0.3
0
0
50
100
150
200
Harmonic order
(b) G ff ( s ) = 1 , Lg = 3L2
Fig. 9.
Simulations under weak and stiff grid conditions
The integral design of a grid-connected inverter (GCI)
control and its LCL filter is proposed in this paper. First, the
capacitor is removed from the LCL filer and the design starts
from a GCI with an L filter and a PI controller. Then, the
capacitor returns to the filter and introduces a resonance, which
is identified to accommodate the preset PI controller by
identifying the phase margin of three cross over frequencies.
Furthermore, the performance under distorted and weak grid
conditions is ensured by the unity feed-forward of the grid
voltage. Moreover, the feed-forward control of the grid voltage
results in an inherent damping effect on the resonance. Hence,
there is no need for any kind of active damping control. In such
an easy way, the controller and filter design are set as a whole
system, so that the control merits of the L-type GCI and the
harmonic attenuation merits offered by the LCL filter are well
integrated. The good dynamics, power quality, and strong
robustness in terms of grid impedance variations are verified
by simulation and experimental results. This shows that the
proposed design is a promising tool for distributed generation
systems.
Co-design of LCL Filter and Control for Grid-Connected Inverters
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10
Journal of Power Electronics, Vol. 14, No. X, July 2014
Yu Zhang was born in Jiangsu Province,
China. He received his B.E., M.E. and Ph.D.
degrees in Electrical Engineering from the
Huazhong University of Science and
Technology (HUST), Wuhan, China, in 1992,
1995 and 2005, respectively. From 1995 to
2002, he was an Engineer working on power
supply applications in a telecommunication company. He is
currently an Associate Professor in the School of Electrical and
Electronic Engineering, HUST, where he teaches power
electronics. His current research interests include power electronics
modeling and control, parallel UPSs, and renewable energy
generation. He has developed several power systems, such as
UPSs, and achieved 4 Scientific and Technological Progress
Awards from the Chinese government and various academic
societies. He is the author of more than 60 technical papers and is
now a Member of the UPS Standard Committee of China.
Mingyu Xue was born in Zhejiang Province,
China, in 1984. He received his B.S. degree in
Electrical Engineering and Automation from
the Dalian University of Technology (DLUT),
Dalian, China, in 2006, and his M.S. degree in
Power Electronics and Drives from the
Huazhong University of Science and
Technology (HUST), Wuhan, China, in 2008. He is currently
working toward his Ph.D. degree at HUST. He is now with the
Shanghai United Imaging Healthcare Company. His current
research interests include power electronic converters, renewable
energy and distributed generation.
Minying Li was born in Shanxi Province,
China. He received his B.E. degree in
Electronics Engineering from the Shanxi
Technology College, Xian, China, in 1983. He
joint the Guangdong Zhicheng Champion
Group Company, in 1999, and is currently the
Chief Engineer of this company. He has
developed many types of power systems, such as UPSs. He
received a National Golden Medal of Patent in 2005 and was
awarded as a Distinguish Specialist with a special allowance
from the State Council of China. He is now the secretary general
of the UPS Standard Committee of China.
Yong Kang was born in Hubei Province,
China. He received his B.E., M.E., and Ph.D.
degrees from the Huazhong University of
Science and Technology (HUST), Wuhan,
China, in 1988, 1991 and 1994, respectively.
In 1994, he joined HUST as a Lecturer and
was promoted to Associate Professor, in 1996,
and to Full Professor, in 1998. He is currently
the Vice Chairman of the China UPS Standard Committee and the
Head of the School of Electrical and Electronic Engineering,
HUST. His current research interests include power electronic
converters, ac drivers, renewable energy generation, and EMC
techniques. He is the author of more than 200 technical papers and
is an associate editor of the Journal of Power Electronics.
Josep M. Guerrero received his B.S.
degree
in
Telecommunications
Engineering, his M.S. degree in
Electronics Engineering, and his Ph.D.
degree in Power Electronics from the
Technical University of Catalonia,
Barcelona, Spain, in 1997, 2000 and 2003,
respectively. Since 2011, he has been a
Full Professor with the Department of Energy Technology,
Aalborg University, Denmark, where he is responsible for the
Microgrid Research Program. Since 2012, he has been a Guest
Professor at the Chinese Academy of Science, Beijing, China,
and the Nanjing University of Aeronautics and Astronautics,
Nanjing, China. Since 2014, he has been a Chair Professor at
Shandong University, Jinan, China. His current research interests
include different microgrid aspects, including power electronics,
distributed energy-storage systems, hierarchical and cooperative
control, energy management systems, and the optimization of
microgrids and islanded minigrids. Professor Guerrero is an
Associate Editor for the IEEE Transactions on Power Electronics,
the IEEE Transactions on Industrial Electronics, and the IEEE
Industrial Electronics Magazine. He is also an Editor for the
IEEE Transactions on Smart Grid. He has been a Guest Editor of
the IEEE Transactions on Power Electronics Special Issues:
Power Electronics for Wind Energy Conversion and Power
Electronics for Microgrids; the IEEE Transactions on Industrial
Electronics Special Sections: Uninterruptible Power Supplies
systems, Renewable Energy Systems, Distributed Generation and
Microgrids, and Industrial Applications and Implementation
Issues of the Kalman Filter; and the IEEE Transactions on Smart
Grid Special Issue on Smart DC Distribution Systems. He was
the Chair of the Renewable Energy Systems Technical
Committee of the IEEE Industrial Electronics Society. In 2014,
he was awarded as an ISI Highly Cited Researcher.